Principles of Quantum Mechanics   as Applied to Chemistry and Chemical Physics

Principles of Quantum Mechanics as Applied to Chemistry and Chemical Physics


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\u142\ufffdi \u142i and the function w(x) are always positive. Therefore, we have
Æi ˆ Æ\ufffdi and the eigenvalues Æi are real. For the situation where i 6ˆ j and
Æi 6ˆ Æ\ufffdj , the integral in equation (3.21) must vanish,…
\u142\ufffdj (x)\u142i(x)w(x) dx ˆ 0 (3:22)
Thus, the set of functions \u142i(x) for non-degenerate eigenvalues are mutually
orthogonal when integrated with a weighting function w(x). Eigenfunctions
corresponding to degenerate eigenvalues can be made orthogonal as discussed
earlier.
The discussion above may be generalized to more than one variable. In the
general case, equation (3.18) is replaced by
A^\u142i(q1, q2, . . .) ˆ Æiw(q1, q2, . . .)\u142i(q1, q2, . . .) (3:23)
and equation (3.22) by…
\u142\ufffdj (q1, q2, . . .)\u142i(q1, q2, . . .)w(q1, q2, . . .) dq1 dq2 . . . ˆ 0 (3:24)
Equation (3.18) can also be transformed into the more usual form, equation
(3.5). We first define a set of functions öi(x) as
öi(x) ˆ [w(x)]1=2\u142i(x) ˆ \u142i(x)=u(x) (3:25)
where
u(x) ˆ [w(x)]ÿ1=2 (3:26)
The function u(x) is real because w(x) is always positive and u(x) is positive
because we take the positive square root. If w(x) approaches infinity at any
point within the range of hermiticity of A^ (as x approaches infinity, for
example), then \u142i(x) must approach zero such that the ratio öi(x) approaches
zero. Equation (3.18) is now multiplied by u(x) and \u142i(x) is replaced by
u(x)öi(x)
u(x)A^u(x)öi(x) ˆ Æiw(x)[u(x)]2öi(x)
If we define an operator B^ by the relation B^ ˆ u(x)A^u(x) and apply equation
(3.26), we obtain
B^öi(x) ˆ Æiöi(x)
which has the form of equation (3.5). We observe that
74 General principles of quantum theory
…
\u142\ufffdj A^\u142i dx ˆ
…
ö\ufffdj uA^uöi dx ˆ
…
ö\ufffdj B^öi dx…
(A^\u142 j)
\ufffd\u142i dx ˆ
…
(A^uö j)
\ufffduöi dx ˆ
…
(B^ö j)
\ufffdöi dx
Since A^ is hermitian with respect to the \u142is, the two integrals on the left of
each equation equal each other, from which it follows that…
ö\ufffdj B^öi dx ˆ
…
(B^ö j)
\ufffdöi dx
and B^ is therefore hermitian with respect to the öis.
3.4 Eigenfunction expansions
Consider a set of orthonormal eigenfunctions \u142i of a hermitian operator. Any
arbitrary function f of the same variables as \u142i defined over the same range of
these variables may be expanded in terms of the members of set \u142i
f ˆ
X
i
ai\u142i (3:27)
where the ais are constants. The summation in equation (3.27) converges to the
function f if the set of eigenfunctions is complete. By complete we mean that
no other function g exists with the property that hg j\u142ii ˆ 0 for any value of i,
where g and \u142i are functions of the same variables and are defined over the
same variable range. As a general rule, the eigenfunctions of a hermitian
operator are not only orthogonal, but are also complete. A mathematical
criterion for completeness is presented at the end of this section.
The coefficients ai are evaluated by multiplying (3.27) by the complex
conjugate \u142\ufffdj of one of the eigenfunctions, integrating over the range of the
variables, and noting that the \u142is are orthonormal
h\u142 j j f i ˆ \u142 j
\ufffd\ufffd\ufffd\ufffdX
i
ai\u142i
* +
ˆ
X
i
aih\u142 j j\u142ii ˆ a j
Replacing the dummy index j by i, we have
ai ˆ h\u142i j f i (3:28)
Substitution of equation (3.28) back into (3.27) gives
f ˆ
X
i
h\u142i j f i\u142i (3:29)
3.4 Eigenfunction expansions 75
Completeness
We now evaluate h f j f i in which f and f \ufffd are expanded as in equation (3.27),
with the two independent summations given different dummy indices
h f j f i ˆ
X
j
a j\u142 j
\ufffd\ufffd\ufffd\ufffdX
i
ai\u142i
* +
ˆ
X
j
X
i
a\ufffdj a jh\u142 j j\u142ii ˆ
X
i
jaij2
Without loss of generality we may assume that the function f is normalized, so
that h f j f i ˆ 1 and X
i
jaij2 ˆ 1 (3:30)
Equation (3.30) may be used as a criterion for completeness. If an eigenfunc-
tion \u142n with a non-vanishing coefficient an were missing from the summation
in equation (3.27), then the series would still converge, but it would be
incomplete and would therefore not converge to f . The corresponding coeffi-
cient an would be missing from the left-hand side of equation (3.30). Since
each term in the summation in equation (3.30) is positive, the sum without an
would be less than unity. Only if the expansion set \u142i in equation (3.27) is
complete will (3.30) be satisfied.
The completeness criterion can also be expressed in another form. For this
purpose we need to introduce the variables explicitly. For simplicity we assume
first that f is a function of only one variable x. In this case, equation (3.29) is
f (x) ˆ
X
i
…
\u142\ufffdi (x9) f (x9) dx9
\ufffd \ufffd
\u142i(x)
where x9 is the dummy variable of integration. Interchanging the order of
summation and integration gives
f (x) ˆ
… X
i
\u142\ufffdi (x9)\u142i(x)
" #
f (x9) dx9
Thus, the summation is equal to the Dirac delta function (see Appendix C)X
i
\u142\ufffdi (x9)\u142i(x) ˆ ä(xÿ x9) (3:31)
This expression, known as the completeness relation and sometimes as the
closure relation, is valid only if the set of eigenfunctions is complete, and may
be used as a mathematical test for completeness. Notice that the completeness
relation (3.31) is not related to the choice of the arbitrary function f , whereas
the criterion (3.30) is related.
The completeness relation for the multi-variable case is slightly more
complex. When expressed explicitly in terms of its variables, equation (3.29) is
76 General principles of quantum theory
f (q1, q2, . . .) ˆ
X
i
…
\u142\ufffdi (q91, q92, . . .) f (q91, q92, . . .)w(q91, q92, . . .) dq91, dq92, . . .
\ufffd \ufffd
3 \u142i(q1, q2, . . .)
Interchanging the order of summation and integration gives
f (q1, q2, . . .) ˆ
… X
i
\u142\ufffdi (q91, q92, . . .)\u142i(q1, q2, . . .)
" #
3 f (q91, q92, . . .)w(q91, q92, . . .) dq1 dq2 . . .
so that the completeness relation takes the form
w(q91, q92, . . .)
X
i
\u142\ufffdi (q91, q92, . . .)\u142i(q1, q2, . . .) ˆ ä(q1 ÿ q91)ä(q2 ÿ q92) . . .
(3:32)
3.5 Simultaneous eigenfunctions
Suppose the members of a complete set of functions \u142i are simultaneously
eigenfunctions of two hermitian operators A^ and B^ with eigenvalues Æi and âi,
respectively
A^\u142i ˆ Æi\u142i
B^\u142i ˆ âi\u142i
If we operate on the first eigenvalue equation with B^ and on the second with A^,
we obtain
B^A^\u142i ˆ ÆiB^\u142i ˆ Æiâi\u142i
A^B^\u142i ˆ âiA^\u142i ˆ Æiâi\u142i
from which it follows that
(A^B^ÿ B^A^)\u142i ˆ [A, B]\u142i ˆ 0
Thus, the functions \u142i are eigenfunctions of the commutator [A^, B^] with
eigenvalues equal to zero. An operator that gives zero when applied to any
member of a complete set of functions is itself zero, so that A^ and B^ commute.
We have just shown that if the operators A^ and B^ have a complete set of
simultaneous eigenfunctions, then A^ and B^ commute.
We now prove the converse, namely, that eigenfunctions of commuting
operators can always be constructed to be simultaneous eigenfunctions.
Suppose that A^\u142i ˆ Æi\u142i and that [A^, B^] ˆ 0. Since A^ and B^ commute, we
have
3.5 Simultaneous eigenfunctions 77
A^B^\u142i ˆ B^A^\u142i ˆ B^(Æi\u142i) ˆ ÆiB^\u142i
Therefore, the function B^\u142i is an eigenfunction of A^ with eigenvalue Æi.
There are now two possibilities; the eigenvalue Æi of A^ is either non-
degenerate or degenerate. If Æi is non-degenerate, then it corresponds to only
one independent eigenfunction \u142i, so that the function B^\u142i is proportional to
\u142i
B^\u142i ˆ âi\u142i
where âi is the proportionality constant and therefore the eigenvalue of B^
corresponding to \u142i. Thus, the function \u142i is a simultaneous eigenfunction of
both A^ and B^.
On the other hand, suppose the eigenvalue Æi is degenerate. For simplicity,
we consider the case of a doubly degenerate eigenvalue Æi; the extension to n-
fold degeneracy is straightforward. The function \u142i is then any linear combina-
tion of two linearly independent, orthonormal eigenfunctions \u142i1 and \u142i2 of A^
corresponding to the eigenvalue Æi
\u142i ˆ c1\u142i1 ‡ c2\u142i2
We